GA-SVR and Pseudo-position-aided GPS/INS Integration ...center.shao.ac.cn › geodesy › publications › TanJin_2015JN.pdf · GA-SVR and Pseudo-position-aided GPS/INS Integration

GA-SVR and Pseudo-position-aidedGPS/INS Integration during

GPS OutageXinglong Tan1, Jian Wang1, Shuanggen Jin2,3 and Xiaolin Meng4

1(School of Environment and Spatial Informatics, China University of Mining andTechnology, Xuzhou, China)

2(Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China)3(Department of Geomatics Engineering, Bulent Ecevit University, Zonguldak, Turkey)4(Institute of Engineering Surveying and Space Geodesy (IESSG), The University of

Nottingham, UK)(E-mail: [email protected])

The performance of Global Positioning System and Inertial Navigation System (GPS/INS)integrated navigation is reduced when GPS is blocked. This paper proposes an algorithm toovercome the condition where GPS is unavailable. Together with a parameter-optimisedGenetic Algorithm (GA), a Support Vector Regression (SVR) algorithm is used to constructthe mapping function between the specific force, angular rate increments of INS measure-ments and the increments of the GPS position. During GPS outages, the real-time pseudo-GPS position is predicted with the mapping function, and the corresponding covariancematrix is estimated by an improved adaptive filtering algorithm. A GPS/INS integrationscheme is demonstrated where the vehicle travels along a straight line and around a curve,with respect to both low-speed-stable and high-speed-unstable navigation platforms. Theresults show that the proposed algorithm provides a better performance when GPS isunavailable.

KEY WORDS

1. GPS/INS integrated navigation. 2. Pseudo-GPS position. 3. Support Vector Regression4. Improved adaptive filtering

Submitted: 8 March 2014. Accepted: 10 January 2015. First published online: 13 February 2015.

1. INTRODUCTION. The Global Positioning System (GPS) and InertialNavigation Systems (INS) have been widely used for navigation. Since GPS satellitesignals are subjected to external environmental disturbance, signal blockage is fre-quently experienced in urban environments, while INS errors accumulate quicklywith time. To provide continuous and reliable navigation solutions, the two systemsare usually employed together for their complementary characteristics. The degra-dation of GPS/INS navigation performance is unavoidable when GPS signals are de-graded in some areas, such as forests and canyons. Special approaches should be

THE JOURNAL OF NAVIGATION (2015), 68, 678–696. © The Royal Institute of Navigation 2015doi:10.1017/S037346331500003X

mailto:[email protected]

developed to overcome this problem, such as GPS integration with additional sensors,e.g., odometers, magnetometers, broadband communication networks (cellular GSM,etc.) or receiver elevation knowledge obtained from a digital terrain model (Danezisand Gikas, 2013; Farrell, 2008; Van Diggelen, 2009).Artificial intelligence and machine learning can also be used to overcome this

problem by taking INS data as the input and the difference from the GPS positionas the output. A multi-sensor system integration approach was introduced to fusedata from an INS and GPS hardware-utilising wavelet multi-resolution analysis andArtificial Neural Networks (ANN) (Chiang et al., 2003; Noureldin et al., 2004). Amulti-layer neural network was trained to map the vehicle dynamics correspondingto Kalman Filter (KF) states, which can be used to correct INS measurements whenGPS measurements are unavailable (Wang et al., 2006). An artificial intelligence-based segmented forward predictor was proposed to update the position and velocityarchitecture by utilising radial basis function neural networks for the purpose of pro-viding metre-level positioning solutions during GPS outages (Semeniuk andNoureldin, 2006). An auxiliary fuzzy-based model was presented for predicting theKF positioning error states during GPS signal outages (Abdel-Hamid et al., 2007).GPS/INS navigation solutions were calculated intelligently using the ANN based onupdating the INS in a Kalman filter structure. To overcome the limitations of multi-layer feed-forward neural networks and KF-based schemes, a constructive neuralnetwork was proposed to improve positioning accuracy by 55% during GPS signaloutages (El-Sheimy et al., 2008; Huang and Chiang, 2008). However, with neural net-works it is difficult to avoid the problems of a local optimal solution, determining thetopological structure and the “curse of dimensionality”.The Support Vector Regression (SVR) algorithm is a new regression technology

based on the Vapnik-Chervonenkis (VC) dimension theory of statistical learning andthe structural risk minimisation principle (Vapnik, 2000). The input is transformedinto a high-dimension characteristic space through a nonlinear transformation,and the optimum linearity regression function is sought. The SVR avoids over-fitting by choosing a specific hyperplane among the feature spaces and overcomesthe major problems faced by typical neural networks, such as local minima, over-fitting or over-training, etc. (Frangos et al., 2010), allowing for a much more hands-off training process that is easily deployable and scalable. Xu et al. (2010) establishedthe regression model between the INS output data and the GPS position differencebased on the least squares support vector machine, and the results show that theleast squares support vector machine is superior to the neural network algorithm.However, the method of acquiring the optimal parameters for SVR model trainingwas not provided.In this paper, the pseudo-GPS position-aided GPS/INS navigation is presented

during GPS outages. With the optimal parameters from genetic algorithms, theSVR algorithm is used to train a regression model between the specific force andangular rate increments of the INS measurements as input and the increments ofGPS position as output. The pseudo-GPS position is calculated with the regressionmodel. An improved adaptive filtering algorithm is then designed to estimate thecovariance matrix of the pseudo-GPS position in real-time. Finally, the pseudo-position-aided navigation solution is given. The structure of this paper is as follows:Section 2 briefly introduces the theory and model of GPS position-aided INS and

679GA-SVR AND PSEUDO-POSITION-AIDED GPS/ INSNO. 4

INS mechanisation equations, Section 3 presents the Genetic Algorithm-SupportVector Regression (GA-SVR) model and its training parameters as well as an optimi-sation method based on GA. The calculation of the pseudo-position and covarianceestimation scheme based on improved adaptive filtering are given in Section 4.Section 5 shows testing results of the pseudo-position-aided low-cost INS scheme,and the conclusions are given in Section 6.

2. CONVENTIONAL GPS/INS INTEGRATED NAVIGATION. A LooselyCoupled (LC) Extended Kalman Filtering (EKF) (Falco et al., 2012; Faruqi andTurner, 2000) is applied for GPS/INS integration, with the states as follows:

X¼½δrN; δrE; δrD; δvN; δvE; δvD; δfRo; δfPi; δfHa;∇bx;∇by;∇bz;

∇fx;∇fy;∇fz; εbx; εby; εbz; εfx; εfy; εfz; δlN; δlE; δlD; δgN; δgE; δgD�Tð1Þ

where δrN, δrE, δrD are the position errors, δvN, δvE, δvD are the velocity errors, δϕRo,δϕPi, δϕHa are the attitude errors, ∇bx, ∇by, ∇bz are the accelerometer biases, ∇fx, ∇fy,∇fz are the accelerometer scale factor errors, εbx, εby, εbz are the gyro drifts, εfx, εfy,εfz are the gyro scale factor errors δlN, δlE, δlD represent the GPS antenna to INSlever arm measurement errors and δgN, δgE, δgD denote the gravity uncertainty errors.For the GPS/INS integrated navigation system, the discrete-time form of the

dynamic model is given as:

Xk ¼ Φk;k�1Xk�1 þ wk ð2Þwhere xk and xk–1 are the state vector at epoch k and k–1, respectively,Φk,k–1 is the statetransition matrix from epoch k–1 to k, and wk is the dynamic process noise. The dif-ference in position between GPS measurements and INS measurements in theNorth-East-Down (NED) frame is regarded as measurements, and the observationequation can be written as:

Zk¼ HkXkþrk ð3Þwhere zk denotes the difference between INS position and GPS position at epochrk,Hk= [I3×3, 03 × 24] is the observation matrix, and rk represents the measurement noise.We assume wk and rk satisfy the following conditions:

EðwkÞ ¼ 0;EðwkwTj Þ ¼ Qkδk;j

EðrkÞ ¼ 0;EðrkrTj Þ ¼ Rkδk;j

EðwkrTj Þ ¼ 0

8><>: ð4Þ

where δk,j is the Kronecker δ-function, Qk is the variance-covariance matrices of states,and Rk is the variance-covariance matrices of measurements. The discrete-timeKalman filter is then given by the following equations.Prediction stage:

X̂ k;k�1 ¼ Φk;k�1X̂ k�1 ð5Þ

Pk;k�1 ¼ Φk;k�1Pk�1ΦTk;k�1þQk�1 ð6Þ

680 XINGLONG TAN AND OTHERS VOL. 68

Update stage:

Kk¼ Pk;k�1HTk HkPk;k�1HT

k þ Rk� ��1 ð7Þ

X̂ k¼ X̂ k;k�1 þ Kk Zk�HkX̂k;k�1� � ð8Þ

Pk ¼ I � KkHk½ �Pk;k�1 ð9Þwhere X̂ k;k�1 is the a priori state estimate, Pk,k–1 is the a priori estimate variance-covari-ance, Kk is the Kalman matrix, X̂ k is the a posteriori state, and Pk is the a posterioriestimate variance-covariance (Bar-Shalom et al., 2001; Chiang et al., 2012; Gikaset al., 1995; Parnian and Golnaraghi, 2010).

3. SVR MODELTRAINING BASED ON GA. When the vehicle moves steadilywithout significantly drastic changes, there is a relatively high correlation between theincrements of the GPS position and the specific force and angular rate increments ofthe INS measurements (Wang et al., 2006; Xu et al., 2010). The SVR model of corre-lation is trained based on the GA in this paper, with the specific force and angular rateincrements of the INS measurements as the input X and the corresponding incrementsof the GPS position as the output Yof the SVR. The GA is used to search the optimalparameters of the SVR before training. A detailed block diagram of the algorithm isshown in Figure 1.

3.1. Principle of Support Vector Regression. The SVR algorithm seeks therelationship between the input and output for a training set of data (xi, yi), i = 1,2,…,l, xi ϵR

n,yi ϵR, where xi is the ith input and yi is the corresponding output. The SVRmodel for nonlinear function estimation has the following representation in thefeature space:

f ðxÞ¼ωTφðxÞþb ð10Þwhere the term ɷ is the weight vector. The nonlinear function φ(x) maps the input xspace to a higher dimensional feature space. The term b is the bias term.

Figure 1. GA-SVR Model training.


It is assumed that ε is the maximum residual between output y and the theoreticalvalue of f(x), so

�ε< y� f ðxÞ<ε ð11ÞTo obtain the optimum value εmin among ε

minω;b

ε

s:t:� ε< y� f ðxÞ< ε; i ¼ 1; 2; :::; lð12Þ

with slack variable ξð�Þ ¼ ðξ1; ξ�1; :::; ξl; ξ�l Þ � 0 and penalty parameter C added toEquation (12), the calculation of ω and b can be altered to the optimisation problemas Equation (13) (Gunn, 1998):

minω∈Rn;b∈R;ξð�Þ∈R2l

τðω; ξð�ÞÞ ¼ 12jjωjj2 þ C

l

Xl

i¼1

ξi þ ξ�i� � ð13Þ

To solve the optimisation problem above, the Lagrangian function is constructed:

L ω; b; ξ; αð Þ ¼ 12

ωj jj j2þCl

Xl

i¼1

ξi þ ξ�i� ��Xl

i¼1

ηiξi þ η�i ξ�i

� �

�Pli¼1

αi εþ yi � ω xð Þ � bð Þ �Pli¼1

α�i ε� yi þ ω xð Þ þ bð Þð14Þ

where α(*) are the Lagrange multipliers αð�Þ ¼ α1; α�1; . . . ; αl; α�l

� �T� 0. According tothe Wolfe duality theory (Wolfe, 1961), the conditions for optimality are given by:

∂L∂ω

¼ ω�Xl

i¼1

α�i � αi� �

xi ¼ 0

∂L∂b

¼Xl

i¼1

αi � α�i� � ¼ 0

∂L∂ξi

¼ Cl� αi � ηi ¼ 0

∂L∂ξ�i

¼ Cl� α�i � η�i ¼ 0

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð15Þ

Substituting Equation (15) into Equation (14), Equation (13) can then be expressed asEquation (16):

minαð�Þ∈R2l

12

Xl

i; j¼ 1

α�i � αi� �

α�j � αj� �

φTðxiÞφðxjÞ þ εXl

i¼1

α�i þ αi� ��Xl

i¼1

yi α�i � αi� �

s:t:Xl

i¼1

α�i � αi� � ¼ 0; 0 � α�i ; αi �

Cl; i ¼ 1; 2 . . . ; l

ð16Þ

Equation (16) belongs to the convex quadratic programming problem, and the feasible

region is empty, meaning that the optimal solution of �α ¼ �α1; �α�1; . . . ; �αl; �α�l

� �Tin


Equation (16) is solved. ω and b can be calculated as follows (Berk, 2008; Joachims,2002; Williams, 2011):

ω ¼Xl

i¼1

�α�i � �αi� �

φðxiÞ

b ¼ yi � ωφðxiÞ � ε; 0< �α�i < C

b ¼ yi � ωφðxiÞ þ ε; 0< �αi < C

� ð17Þ

As a result, the SVR model for nonlinear function estimation becomes:

f xð Þ ¼Xl

i¼1

�α�i � �αi� �

K xi; xð Þ þ b ð18Þ

where K(xi, x) = φT( xi) φ( xj) is a positive definite kernel matrix. Note that the RadialBasis Function (RBF) has an advantage in processing linearly inseparable data, andtherefore the RBF kernel (K(xi, xj) = exp(−γ||xi− xj||

2)) is chosen as the kernel func-tion. The γ is the kernel width: small and kernel width may cause over-fitting, and alarge kernel width may cause under-fitting (Chang et al., 2005). A small penalty par-ameter (C) leads to over-fitting and a large one brings about under-fitting (Alpaydin,2004). The performance of SVR with Gaussian RBF kernel is sensitive to the kernelwidth (γ) and penalty parameter (C). Several methods can be used to obtain theoptimal γ and C, e.g., bootstrapping, VC bounds statistical learning theory, and infer-ence or Bayesian learning methods (Cristianini and Ricci, 2008; Kecman, 2005).Genetic algorithms are developed in this paper, shown in the next section.

3.2. Parameter optimisation based on genetic algorithms. Genetic algorithms area family of computational models inspired by evolution. These algorithms encode apotential solution to a specific problem on a simple chromosome-like data structure,and they apply recombination operators to these structures in a way that preservescritical information (Goldberg and Holland, 1988). With respect to ϒ, C of SVR,the solutions of the parameter optimisation problem can be expressed as follows.Step 1: Encoding. Note that there is only one change between two adjacent numbers

and the grey code is developed in this paper. The relationship between binary code Band grey code G is:

Bn�1 ¼ Gn�1

Gi ¼ Giþ1 ⊕ Gi; i ¼ 0; 1; 2; . . . ; n� 2

�ð19Þ

where ⊕ represents the XOR operator.Step 2: Initialisation. Set the range of parameters 0≤ϒ≤ 1000, 0 <C≤ 500, 20

chromosomes of each parameter γ, C are generated randomly and the maximalgenetic generation is 200.Step 3: Fitness calculation of the individual. The fitness function is the basis of the

optimisation to evaluate the quality of the individual. The Root Mean Square (RMS)of the SVR-trained residual is calculated based on K-fold cross-validation with the 20chromosomes. Descending through the chromosomes according to the RMS, thefitness of each chromosome is then shown in Equation (20):

FitnV Posð Þ ¼ 2� spþ 2 × ðsp� 1Þ × Pos� 1N � 1

ð20Þ


where sp is the assigned press difference, Pos is the position of the chromosomes and Nis the population size. FitnVϵ[1, 2].Step 4: Genetic operators.Selection: The population of the next generation is formed by means of a probabil-

istic reproduction process. Individuals with a higher fitness usually have a greaterchance for the next generation. The selected probability Psi of the ith chromosomexi is shown in Equation (21).

Psi ¼ fi.XN

j¼1fj ð21Þ

whereN denotes the size of the population and fi the fitness function of chromosome xi.Crossover: Crossing over tends to enable the evolutionary process to move toward

promising regions of the search space. The next generation is formed between twoselected individuals, called parents, by exchanging parts of their strings. Single-pointcrossover is developed with the probability of 0·7 in this paper as seen in Figure 2.Mutation: Mutation is used to search for additional problem space and to avoid the

local convergence of GA. For each bit in the population in this paper, ‘mutate’ changesthe bit value with a low probability of 0·05.Step 5: End the GA procedure, and output the optimal chromosome if the genetic

generation reaches the maximum value, else, go to step 3.

4. PSEUDO-GPS POSITION-AIDED INS NAVIGATION. When GPS signalsare available, the LC strategy, including an EKF, is adopted to combine the GPSand INS values to estimate navigation solutions. The INS errors in 24 states estimatedby EKF correct the INS model in real-time. Simultaneously, the regression model istrained based on the GA-SVR, which maps the increments of the GPS positionwith the specific force and angular rate increments of the INS measurements. If theGPS signals are unavailable, the pseudo-GPS position can be estimated based onthe trained model and the specific force and angular rate increments of the INS mea-surements. An improved adaptive filtering is designed to estimate the covariancematrix of the pseudo-GPS positions in real time. The INS errors estimated byimproved adaptive filtering correct the INS model. The pseudo-GPS position-aidednavigation solutions are then calculated as shown in Figure 3.

4.1. Pseudo-GPS position calculation. Assume that ti is the epoch before GPSbecomes unavailable, Pi is the corresponding position of the GPS. Δt is the samplinginterval of the GPS measurements. At the next epoch ti+ 1 (ti+ 1 = ti + Δt), the GPS po-sition incrementsΔPj canbederivedusing the trainedGA-SVRmodel, specific force incre-ments ∫

tiþ1

ti f bibdt and angular rate increments ∫tiþ1

ti ωbibdt (according to Equation (18)).

Figure 2. Single-point crossover.


The pseudo-GPS position at epoch ti+1 can then be obtained as:

Pj ¼ Pi þ ΔPj ð22ÞAfter n intervals, the final pseudo-GPS position at epoch tk (tk= ti+ nΔt) is obtained as:

Pk ¼ Pi þXnt¼1

ΔPt ð23Þ

4.2. Improved adaptive filtering. The EKF is disabled due to the absence of thecovariance matrix of the pseudo-GPS position from the GA-SVR algorithm. In thispaper, an improved adaptive filtering algorithm is proposed by combining Sage-Husa Adaptive Filtering (SHAF) with robust filtering. The SHAF can estimate thecovariance matrix in real time according to the innovation to improve the estimationaccuracy (Ding et al., 2007). The predicted pseudo-GPS positions inevitably containbig errors/biases, so that a robust algorithm which can detect and solve the errors isneeded, such as the equivalent weight method (Yuanxi, 1994) or ReceiverAutonomous Integrity Monitoring (RAIM) (Hewitson and Wang, 2007, 2010).

. Sage-Husa adaptive filtering

The innovation sequence is defined as Equation (24):

vk ¼ Zk �HkX̂k;k�1 ð24ÞThe predicted error covariance matrix from the innovation sequence is:

EðvkvTk Þ ¼ HkPk;k�1Hk þ Rk ð25ÞThere is clearly a relationship in Equation (25) to estimate Rk. However, it requires alimited number (called ‘estimation window size’) of innovation samples to calculateE(vkvk

T). Considering the number of pseudo-measurements, we use both a priori knowl-edge Rk−1 and innovation vkvk

T to estimate the covariance matrix Rk as follows

Figure 3. GA-SVR-Based Pseudo-Position-aided INS Navigation algorithm.


(Lu et al., 2007; Sage and Husa, 1969):

Rk ¼ 1� dkð ÞRk�1 þ dkðvkvTk �HkPk;k�1HTk Þ ð26Þ

where dk ¼ 1� e1� ek

; 0< e< 1, e is the forgetting factor.

dk ¼ 1� e1� ek

; 0< e< 1 ð27Þ

. Robust filtering

The residual sequence is defined as:

εk ¼ Zk �HkX̂k ð28ÞThen the mean square error factor σ̂0 is calculated with the median method as:

σ̂0 ¼ medifj

ffiffiffiffiffiPi

pεijg=0:6745 ð29Þ

where εi is the ith element of the residual sequence with the weightPi. The standardisedresidual si of εi as:

si ¼ jffiffiffiffiffiPi

pεij=σ̂0 ð30Þ

Robust factors γi based on IGGIII weight function (Yuanxi, 1994) are constructed as:

γi ¼1; si � k0k0si×

k1 � sik1 � k0

�2; k0 < si � k1

10�30; si > k1

8>>><>>>:

ð31Þ

where k0, k1 are threshold value and k0 = 1.0 ~ 1.5, k1 = 2.5 ~ 8.0.If si≤ k0,we think that the ith pseudo-GPS position has no error; if k0 < si≤ k1, we

think that the ith pseudo-GPS position has small error; and if si > k1, we think that theith pseudo-GPS position has big error. To reduce the impact of the big errors/biases tothe navigation solutions, the covariance matrix of pseudo-GPS positions is amplifiedwith robust factors as follows:

R̂i ¼ R̂i=γi ð32Þ

5. TEST RESULTS. Two sets of Leica 1200 Base and Rover GPS Systems andSPAN-CPT INS units are used. Firstly initial alignment is done with the supportingsoftware of SPAN-CPT, and then the raw IMU data from SPAN-CPT and GPSdata from Leica are collected to validate the proposed algorithm in this paper. Thesensor specifications of the SPAN-CPT are listed in Table 1.If GPS signals are available, the loosely coupled strategy is adopted to calculate the

navigation solutions of the GPS/INS integrated system based on EKF. Figure 4ashows the position errors of the GPS/INS navigation solution for the latitude, longi-tude and height when GPS signals are available. The lever arm between GPSantenna phase centre and IMU navigation centre directly affects the output position


in a GPS/INS integrated system. According to the algorithm (Geng et al., 2011; Tanget al., 2009), Figure 4b shows the estimated lever arm, the red line represents truevalues.The feasibility of the algorithm is verified using three tests. The data were recorded

for post processing. Test 1 moved along a straight line with respect to a low-speed-stable navigation platform, Test 2 moved along a straight line with respect to ahigh-speed-unstable navigation platform, and Test 3 moved along a curve.Trajectories of these three tests can be seen in Figure 5.Test 1: 4100 seconds of RTK-GPS (1 Hz) and IMU (100 Hz) data were collected

when the vehicle moved at a speed of 20 km/h. Navigation solutions between the1580th and 1700th seconds are provided with the proposed algorithm, assumingthat the GPS signal was unavailable during that time. With a similar motion state,101 groups of data from the 1380th to 1480th seconds were chosen as the GA-SVRtraining samples. The data from 1481st to 1579th seconds were not used because thevehicle was forced to stop at a crossroad.Test 2: 2100 seconds of RTK-GPS (1 Hz) and IMU (100 Hz) data were measured at

a speed of 70 km/h. Navigation solutions between the 561st and 800th seconds weresolved with the algorithm, assuming that the GPS signal was unavailable duringthat time. With a similar motion state, 201 groups of data from the 360th to the560th seconds were chosen as the GA-SVR training samples.Test 3: A set of data was selected when the vehicle moved along a curve at a speed of

20 km/h. Navigation solutions between the 943rd and 1023rd seconds are provided

Table 1. INS technical specifications.

Gyro Accelerometer

Range ±375°/s ±10 gBias 20°/hr 50 mgBias Stability ±1°/hr ±0·75 mgScale Factor 1500 ppm 4000 ppmRandom Walk 0·0667°/√hr 60 μg/√Hz

Figure 4. (a) Navigation solution errors (Left). (b) Lever arm estimation (Right).


with the proposed algorithm, assuming that the GPS signal was unavailable duringthat time. With a similar motion state, 81 groups of data from the 313rd to the393rd seconds were chosen as the GA-SVR training samples.

5.1. GPS position increments based on the GA-SVR. Figure 6 shows the trainingdata of the three tests. Note that the specific force and angular rate increments of theINS measurements are multiplied by the sample interval 0·01 s. In Test 1 and Test 2,the roll and pitch of the angular rate are no more than 0·005 radians in magnitude,and the heading is no more than 0·02 radians, which illustrates that the direction ofmovement of the test vehicle remains stable. The specific force increments in Test 1appear smoother than those in Test 2, and the result for the change of the GPS positionincrements in Test 1 is more stable than that for Test 2. From the increments ofheading, latitude and longitude, it is obvious that Test 3 passed through a curve.

Figure 5. Trajectories for trained and predicted data. (a) Test 1 (Left). (b) Test 2 (Middle). (c) Test 3(Right).

Figure 6. GA-SVR training data of three tests.


Figure 7 shows the process of seeking the optimal parameters γ and C in an SVRbased on genetic algorithms. The iteration-stopping criterion is defined as a differencebetween two adjacent fitness levels of less than 0·001. The iterations in latitude, longi-tude, and height are: 57, 35, and 9, respectively, in Test 1; 193, 130, and 2, respectively,in Test 2; 13, 32, and 2, respectively, in Test 3. A faster convergence rate is achieved inthe height direction for tests, and the slowest convergence rate arises in the latitude inTest 2, where the changes in the GPS position increments are the largest. The optimalparameters γ and C can be seen in Table 2.Based on SVR theory, the sample data are trained to construct regression models

with the optimal parameters γ and C in the SVR determined by genetic algorithms.Figure 8 shows the results and deviations of trained and predicted GPS position

Figure 7. Genetic algorithm fitness curves. (a) Test 1 (Top). (b) Test 2 (Middle). (c) Test 3 (Bottom).

Table 2. Results of GA-SVR.

Test Direction

Results of GA Numberof supportvectors Bias

RMS error oftrained (m)

RMS error ofpredicted (m)C γ

latitude 0·576 22·020 101 −0·463 7·46e-3 0·0831 longitude 3·194 29·592 101 −0·551 3·44e-4 0·111

height 7·750 107·273 101 −0·456 5·26e-5 0·024latitude 312·228 0·007 191 −0·173 0·994 0·835

2 longitude 354·014 0·001 192 −1·371 0·647 0·515height 302·118 0·012 187 3·501 0·003 0·005latitude 22·254 116·920 77 −0·511 0·163 0·295

3 longitude 6·792 141·278 80 −0·384 0·113 0·201height 51·796 780·042 64 −0·830 0·166 0·017


Figure 8. GA-SVR training results of Latitude, Longitude, and Height. (a) Test 1 (Top). (b) Test 2(Middle). (c) Test 3 (Bottom).


increments based on the GA-SVR algorithm. With a more smooth and stable state,the trained and predicted accuracy in Test 1 and Test 3 is much better than that inTest 2. The RMS of the errors of the three tests is shown in Table 2.

5.2. Pseudo-GPS position-aided navigation. The pseudo-GPS position is calcu-lated by adding the predicted GPS position increments to the GPS position (recordedbefore the GPS outage), as shown in Section 4.1. Figure 9 shows the pseudo-GPS po-sition and its deviations for the three tests. Note that latitude deviations are transformedto metres by multiplying the radius of the curvature in the meridian, and longitudedeviations are transformed by multiplying the radius of the curvature in the prime ver-tical and cosine of the latitude. The RMS of the latitude, longitude, and height devia-tions are 1·440 m, 0·717 m, and 0·561 m, respectively in Test 1, 11·641 m, 20·148 m, and4·350 m, respectively in Test 2, and 1·576 m, 1·874 m, and 0·077 m, respectively in Test3. The accuracy of the pseudo-GPS position in Test 1 and Test 3 is obviously muchhigher than that in Test 2 due to low speeds and smooth operation.INS/RTK-GPS stands for the conventional GPS/INS loosely coupled integration al-

gorithm when GPS signals are available; INS/GA-SVR means improved adaptive fil-tering with the pseudo-GPS positions during the absence of GPS signals; INS-onlyrepresents the navigation system depending solely on the equipped INS. Figure 10shows deviations in the comparisons of the INS/RTK-GPS, INS/GA-SVR, andINS-only algorithms in three dimensions. The results indicate that deviations forINS-only are drifted to 27 m in 120 seconds when the GPS is unavailable, but theRMS of the deviation with INS/GA-SVR is 1·699 m with a maximum deviation ofno more than 2·734 m in Test 1. The performance is very stable. In Test 2, the devia-tions in the INS-only drift to 119 m in 240 seconds, while the RMS of the deviationwith INS/GA-SVR is 24·026 m, with a maximum deviation of less than 36·403 m.Navigation solutions undulate frequently from 560∼680 seconds and are better than

Figure 9. Pseudo-GPS position comparison and deviations. (a) Test 1 (Left). (b) Test 2 (Middle).(c) Test 3 (Right).


INS-only after that point as the result of the accuracy of the pseudo-GPS position. InTest 3, deviations in the INS-only drift to 9 m in 81 seconds, while the RMS of the de-viation with INS/GA-SVR is 2·472 m, with a maximum deviation of no more than3·600 m.Figure 11 shows the velocity comparison for the INS/RTK-GPS and INS/GA-SVR

algorithms in the north, east, and up directions. In Test 1, the RMS of the deviationusing the INS/GA-SVR algorithm are 0·051 m/s, 0·068 m/s, and 0·015 m/s, with themaximum no more than 0·146 m/s, 0·187 m/s, and 0·041 m/s in the north, east, and

Figure 10. Position errors comparison in three dimensions. (a) Test 1 (Top). (b) Test 2 (Middle).(c) Test 3 (Bottom).


up directions, respectively. This is almost identical to the true velocity with INS/RTK-GPS. In Test 2, the RMS of the deviation using the INS/GA-SVR algorithm are 0·560m/s, 0·406 m/s, and 0·075 m/s, with a maximum of no more than 1·003 m/s, 0·782 m/s,and 0·226 m/s in north, east, and up directions, respectively. In Test 3, the RMS of thedeviation using the INS/GA-SVR algorithm are 0·104 m/s, 0·077 m/s, and 0·033 m/s,with a maximum deviation of less than 0·239 m/s, 0·187 m/s, and 0·040 m/s in north,east, and up directions, respectively. This result indicates that the velocity in the up di-rection is close to that with INS/RTK-GPS, but there are small deviations betweenINS/RTK-GPS and INS/GA-SVR in the north and east directions.Figure 12 shows an attitude comparison of the INS/RTK-GPS and INS/GA-SVR

algorithms in the roll, pitch, and heading. In Test 1, the RMS of the deviation withthe INS/GA-SVR algorithm are 0·109°, 0·250°, and 0·222°, with the maximum nomore than 0·161°, 0·342°, and 0·290° in the roll, pitch, and head, respectively, whichare almost identical to the attitude of INS/RTK-GPS. In Test 2, the RMS of the devi-ation with the INS/GA-SVR algorithm are 0·039°, 0·135°, and 0·985°, with amaximum of no more than 0·088°, 0·278°, and 1·509° in the roll, pitch, and head, re-spectively. In Test 3, the RMS of the deviation with the INS/GA-SVR algorithm are0·023°, 0·040°, and 0·111°, with a maximum of no more than 0·058°, 0·070°, and0·168° in the roll, pitch, and head, respectively. These results indicate that the attitudein the roll and pitch is close to INS/RTK-GPS, but there are small differences betweenINS/RTK-GPS and INS/GA-SVR in the heading. The statistical results for the vel-ocity and attitude are given in Table 3.The results indicate that 1) Based on the GA-SVR algorithm, the accuracy of the

pseudo-GPS position is high when the vehicle operates a low-speed-stable navigationplatform along both a straight line and a curve, and low if the vehicle operates ahigh-speed-unstable navigation platform; 2) The velocity and attitude of navigationsolutions are very close to the true value, and they benefit from the implementation

Figure 11. Velocity comparison. (a) Test 1 (Top). (b) Test 2 (Middle). (c) Test 3 (Bottom).


of improved adaptive filtering that only adjusts the position covariance matrix ratherthan other states; and 3) The accuracy of navigation solutions depends largely upon theaccuracy of the pseudo-GPS position when the GPS is unavailable.

6. CONCLUSIONS. To overcome the shortcomings of GPS/INS integrated navi-gation during GPS outages, we have proposed a pseudo-position-aided INS navigationalgorithm. Based on GA-SVR, the algorithm uses the predicted pseudo-position andimproved adaptive filtering to calculate reliable navigation solutions. The proposed al-gorithm has been tested on low-speed-stable and high-speed-unstable navigation plat-forms where the vehicle travels along a straight line and around a curve. Results showthat the proposed new approach can provide reliable and accurate navigation solutionswhen the GPS is unavailable. The calculation of the optimal parameters in an SVRtraining model will take more time, which will affect the real-time navigation of theintegrated system. Thus the optimal parameters should be trained in advance accord-ing to the INS performance and the different motion states of the vehicle. With the

Figure 12. Attitude comparison. (a) Test 1 (Top). (b) Test 2 (Middle). (c) Test 3 (Bottom).

Table 3. Results of velocity and attitude comparison.

Test

Velocity deviation (m/s) Attitude deviation (degree)

North East Up Roll Pitch Head

1 Max 0·146 0·187 0·041 0·161 0·342 0·290RMS 0·051 0·068 0·015 0·109 0·250 0·222

2 Max 1·003 0·782 0·226 0·088 0·278 1·509RMS 0·560 0·406 0·075 0·039 0·135 0·985

3 Max 0·239 0·187 0·040 0·058 0·070 0·168RMS 0·104 0·077 0·033 0·023 0·040 0·111


further development of parallel computing technology and computer performance,this issue will be resolved in the future.

ACKNOWLEDGMENTS

The work is partially sponsored by the Program for New Century Excellent Talents in University(grant number: NCET-13–1019), partially sponsored by the Fundamental Research Funds forthe Central Universities (grant number: 2013RC16); and partially sponsored by A ProjectFunded by the Priority Academic Program Development of Jiangsu Higher EducationInstitutions.

REFERENCES

Abdel-Hamid, W., Noureldin, A. and El-Sheimy, N. (2007). Adaptive fuzzy prediction of low-cost inertial-based positioning errors. Fuzzy Systems, IEEE Transactions on, 15, 519–529.

Alpaydin, E. (2004). Introduction to machine learning. MIT press.Bar-Shalom, Y., Li, X.R. and Kirubarajan, T. (2001).Estimation with applications to tracking and navigation.John Wiley & Sons.

Berk, R. (2008). Support Vector Machines. Springer, New York.Chang, Q., Chen, Q. and Wang, X. (2005). Scaling Gaussian RBF kernel width to improve SVM classifi-cation. Proceedings of the Neural Networks and Brain. ICNN&B’05. International Conference on,

Chiang, K.-W., Duong, T.T., Liao, J.-K., Lai, Y.-C., Chang, C.-C., Cai, J.-M. and Huang, S.-C. (2012). On-Line Smoothing for an Integrated Navigation System with Low-CostMEMS Inertial Sensors. Sensors, 12,17372–17389.

Chiang, K.W., Noureldin, A. and El-Sheimy, N. (2003). Multisensor integration using neuron computing forland-vehicle navigation. GPS solutions, 6, 209–218.

Cristianini, N. and Ricci, E. (2008). Support Vector Machines. Springer US.Danezis, C. and Gikas, V. (2013). An Iterative LiDAR DEM-Aided Algorithm for GNSS Positioning inObstructed/Rapidly Undulating Environments. Advances in Space Research, 52, 865–878.

Ding, W., Wang, J., Rizos, C. and Kinlyside, D. (2007). Improving adaptive Kalman estimation in GPS/INSintegration. Journal of Navigation, 60, 517–529.

El-Sheimy, N., Chiang, K. and Noureldin, A. (2008). Developing a low cost MEMS IMU/GPS integrationscheme using constructive neural networks. IEEE Trans. Aerosp. Electron. Syst, 44, 582–594.

Falco, G., Einicke, G., Malos, J. and Dovis, F. (2012). Performance Analysis of Constrained LooselyCoupled GPS/INS Integration Solutions. Sensors, 12, 15983–16007.

Farrell, J. (2008). Aided navigation: GPS with high rate sensors. McGraw-Hill, New York.Faruqi, F.A. and Turner, K.J. (2000). Extended Kalman filter synthesis for integrated global positioning/inertial navigation systems. Applied mathematics and computation, 115, 213–227.

Frangos, K., Kealy, A., Gikas, V. and Hasnur, A. (2010). Dynamic modeling for land mobile navigationusing low-cost inertial sensors and least squares support vector machine learning. Proceedings of theProceedings of the 23rd International Technical Meeting of The Satellite Division of the Institute ofNavigation (ION GNSS 2010).

Geng, Y., Deurloo, R. and Bastos, L. (2011). Hybrid derivative-free extended Kalman filter for unknownlever arm estimation in tightly coupled DGPS/INS integration. GPS solutions, 15, 181–191.

Gikas, V., Cross, P. and Akuamoa, A. (1995). A rigorous and integrated approach to hydrophone and sourcepositioning during multi-streamer offshore seismic exploration. The Hydrographic Journal, 77, 11–24.

Goldberg, D.E. and Holland, J.H. (1988). Genetic algorithms and machine learning. Machine learning, 3,95–99.

Gunn, S.R. (1998). Support vector machines for classification and regression. ISIS technical report, 14,41–46.

Hewitson, S. and Wang, J. (2007). GNSS receiver autonomous integrity monitoring with a dynamic model.Journal of Navigation, 60, 247–263.

Hewitson, S. and Wang, J. (2010). Extended Receiver Autonomous Integrity Monitoring (e RAIM) forGNSS/INS Integration. Journal of Surveying Engineering, 136, 13–22.


Huang, Y.W. and Chiang, K.W. (2008). An intelligent and autonomous MEMS IMU/GPS integrationscheme for low cost land navigation applications. GPS solutions, 12, 135–146.

Joachims, T. (2002). Support Vector Machines. Springer, US.Kecman, V. (2005). Support Vector Machines – An Introduction. Springer, Berlin Heidelberg.Lu, P., Zhao, L. and Chen, Z. (2007). Improved Sage-Husa Adaptive Filtering and Its Application [J].Journal of System Simulation, 15, 3503–3505.

Noureldin, A., Osman, A. and El-Sheimy, N. (2004). A neuro-wavelet method for multi-sensor system inte-gration for vehicular navigation. Measurement science and technology, 15, 404–412.

Parnian, N. and Golnaraghi, F. (2010). Integration of a multi-camera vision system and strapdown inertialnavigation system (SDINS) with a modified kalman filter. Sensors, 10, 5378–5394.

Sage, A. P. and Husa, G.W. Adaptive filtering with unknown prior statistics. (1969). Proceedings of theProceedings of the Joint Automatic Control Conference, Tokyo, Japan, 760–769.

Semeniuk, L. and Noureldin, A. (2006). Bridging GPS outages using neural network estimates of INS po-sition and velocity errors. Measurement science and technology, 17, 2783–2789.

Tang, Y., Wu, Y., Wu, M., Wu, W., Hu, X. and Shen, L. (2009). INS/GPS integration: global observabilityanalysis. Vehicular Technology, IEEE Transactions on, 58, 1129–1142.

Van Diggelen, F. S. T. (2009). A-GPS: Assisted GPS, GNSS, and SBAS. Artech House.Vapnik, V. (2000). The nature of statistical learning theory. Springer.Wang, J. J., Wang, J., Sinclair, D. and Watts, L. (2006). A neural network and Kalman filter hybridapproach for GPS/INS integration. Proceedings of the 12th IAIN Congress & 2006 Int. Symp. on GPS/GNSS, Jeju, Korea,

Williams, G. (2011). Support Vector Machines. Springer, New York.Wolfe, P. (1961). A duality theorem for nonlinear programming. Quarterly of applied mathematics, 19,239–244.

Xu, Z., Li, Y., Rizos, C. and Xu, X. (2010). Novel hybrid of LS-SVM and kalman filter for GPS/INS inte-gration. Journal of Navigation, 63, 289–299.

Yuanxi, Y. (1994). Robust estimation for dependent observations. Manuscripta geodaetica, 19, 10–17.


Documents

GA-SVR and Pseudo-position-aided GPS/INS Integration ...center.shao.ac.cn › geodesy › publications › TanJin_2015JN.pdf · GA-SVR and Pseudo-position-aided GPS/INS Integration